We propose a Bayesian model predictive control (MPC) framework for uncertainty-aware quantum control. The method embeds Bayesian inference directly into the receding-horizon control loop, enabling online identification of uncertain Hamiltonian parameters and the optimization of feedback policies under realistic noise and actuation constraints. Numerical experiments on single-qubit state-transfer tasks show that Bayesian MPC attains faster convergence and higher terminal fidelities than both nominal MPC and open-loop GRAPE under parameter mismatch, while preserving robustness to decoherence and readout noise. Importantly, the approach remains computationally efficient for short prediction horizons, enabling real-time deployment on near-term quantum hardware. These results establish Bayesian MPC as a principled and practical strategy for adaptive, feedback-based quantum control under uncertainty.